The word "persistent" can have either a positive or negative -- good or bad -- connotation, depending on the circumstances of its use and the listener's interpretation. Persistence is sticking to an idea, opinion or course of action despite extreme challenges or difficult odds. A word's connotation refers to the emotion, idea or images it inspires in readers or listeners.
Positive and Negative Connotations
"Persistent" often is used as a positive adjective to describe someone who relentlessly pursues goals and doesn't run when things get tough. However, "persistent" sometimes is used in place of "stubborn" to describe someone who sticks to an opinion or point of view despite the fatigue, disinterest or difference of opinion of the other person.